Welcome to things-that-count.net. This website describes a collection of antique calculators (“collection Calculant”1) and uses it to help develop a historical account of the way humans developed the need and capacity to calculate, the things they used to help them, and how human societies (and even human brains) evolved with those developments.

The account given here is of a 37,000 year story, one in which humans came to count, record their counting and do simple arithmetic sums. Over time these capabilities became essential ingredients in what became increasingly complex societies. More citizens needed to be able to manipulate numbers in their daily lives and, over time, a range of aides of various sorts were developed to assist. At the beginning, the aides were very simple - for example, marks scribed on bones and patterns arranged with pebbles. Later, primitive devices and tables were developed and sold. Over time, much more elaborate mechanical devices were developed to help in this task. Many of these devices, where they survive, now represent little more than mechanical fossils. Unused and largely forgotten their remains are scattered across history from earliest human pre-history to our present moment.

The need for calculation, however has prospered. As societies have become more complex, transactions in them depending on arithmetic (the familiar tasks of counting, adding, subtracting, multiplying and dividing), as well as more complex mathematics, have intensified. Yet over much of this period, for many people in these societies, doing even the simplest arithmetic tasks has been neither easy nor, for some, comprehensible. For this reason finding ways to do these tasks faster and more accurately, and to spread the ability across more people, has been a preoccupation in many societies. It is the approaches that have been taken to aid achieving these simple goals (rather than the development of complex mathematics) which is the primary focus of this website.

Early “calculators” were not things. Rather they were people who were employed to calculate. Over time these people were first aided, but later replaced by calculating devices. These devices became very widely used across many countries. There is evidence we may now be passing the heyday of such stand-alone calculators. This is because, increasingly since the advent of electronic computing, the aides to calculation have begun to appear in virtual form as apps in phones, tablets and laptops. The end of calculators, seen as devices, in this sense is looming.

One might imagine that a history of calculators would consist simply of the progressive discovery and invention of ever more effective and sophisticated calculating devices. Indeed many such accounts do focus on this with loving details of the minutae of mechanical invention. But to focus simply on that is to oversimplify and lose much of what is potentially interesting. Across human history many weird things were indeed devised for doing simple calculations. But the development of these begs a series of questions: When and why were they made, how were they used, and why at times were they forgotten for centuries or even millennia?

The objects in “collection Calculant” described here, which are used to help illustrate answers to these sort of questions, are drawn from across some 4,000 years of history. Each of them was created with a belief that it could assist people in thinking about (and with) numbers. They range from little metal coin-like disks to the earliest electronic pocket calculators - representing a sort of ‘vanishing point’ for all that had come before. The collection and the history it illustrates in a sense form a duet - the two voices each telling part of the story. The history has shaped what has been collected, and the collection has helped shape how the story is told.

1. Initial observations.

As most people know, the spread of electronic personal calculators of the 1970s was followed quickly by the first personal computers. Before long, computer chips began to be embodied in an ever expanding array of converging devices. In turn, ever greater computing power spread across the planet. However sophisticated these modern computers appeared on the outside, and whatever the diversity of functions they performed, at heart they achieved most of this by doing a few things extremely fast. (Central to the things they did were logic operations such as “if”, “and”, “or” and “not”, and the arithmetic operations of addition and subtraction - from which multiplication and division can also be derived). On top of this were layers of sophisticated programming, memory and input and output.

Prior calculating technologies had to rely on slower mechanical processes. This meant they were much more limited in speed, flexibility and adaptability. Nevertheless they too were designed to facilitate the same fundamental arithmetic and logical operations. The technologies of mathematics are in this sense much simpler than the elaborate analytic structures which make up mathematical analysis. And for this reason, it is not necessary to consider all of mathematics in order to follow much of the history of how the technologies to aid mathematical reasoning developed. Just considering the history of the development of aids to calculation can tell a great deal. As already noted, it is that which is dealt with here.

The calculational devices that were developed show an unmistakeable progression in complexity, sophistication and style from the earliest to the latest. Corresponding to this it is possible to construct histories of calculational aids as some sort of evolution based on solving technical problems with consequent improvements in design building one upon the other. But, as already noted, it is also important to understand why they were invented and used.

Fortunately in order to understand what has shaped the development of these calculational aids we can largely avoid talking much about mathematics. This is lucky because mathematics is by now a huge field of knowledge. So you are entitled to relief that in this site we will avoid much of mathematics. We need not touch, for example, on calculus, set and group theory, the mathematics of infinite dimensional vector spaces that make the modern formulation of quantum mechanics possible, and tensors which Einstein used to express his wonderfully neat equations for the shape of space-time.2 It will be sufficient to note that many modern challenges - from the prediction of climate under the stress of global warming, to the simulation of a nuclear reactor accident, to the deconstruction of DNA - could not occur without enormous numbers of calculations which in the end are constituted out of additions and subtractions (and multiplications and divisions) and can only be carried out in workable times with the use of ever faster calculating devices.

Even keeping our attention restricted to the basic arithmetic operations, it turns out we will still encounter some of the curly issues that we would have to think about if we were focussing on the whole evolving field of mathematical thought. Of course history of mathematics is itself a field of scholastic study which can be developed from many perspectives. These include those from the mainstream of history and philosophy of science3 through to the sociology of science.4 Even though this discussion here focuses on only a tiny “arithmetic core” of mathematics it will still be useful to take some account of this literature and its insights. In particular, whether concerning ourselves with the evolution of the simple areas of mathematics or the more obstruse areas, one question is always raised: what led to this particular development happening as and when it did?

2. Did increases in the power of mathematics lead the development of calculators? Was it the other way round?

It might be assumed that arithmetic, and more broadly, mathematics, developed through a process that was entirely internal to itself. For example, this development might have been propelled forward because people could ask questions which arise within mathematics, but require the invention of new mathematics to answer them.

Suppose we know about addition and that 2+2 =4. Then it is possible to ask what number added to 4 would give 2. Answering that involves inventing the idea of a negative number. This leads to progress through ‘completing mathematics’ (i.e. seeking to answer all the questions that arise in mathematics which cannot yet be answered.) That must be part of the story of how mathematics develops. Yet the literature on the history of mathematics tells us this cannot be all.

The idea of ‘mathematics’, and doing it, are themselves inventions. The question of when mathematics might be useful will have different answers in different cultures. Different societies may identify different sorts of issues as interesting or important (and only some of these will be usefully tackled with mathematics). Also different groups of people in those societies will be educated in what is known in mathematics. Finally, different groups of people, or organisations, may have influence in framing the questions that mathematicians are encouraged (and resourced) to explore.

But the same is true of invention. At different times and in different cultures there have been quite different views taken on the value of change, and thus invention. At some points in history the mainstream view has been that the crucial task is to preserve the known truth (for example, as discovered by some earlier civilisation - notably the ancient Greeks, or as stated in a holy book). At other times or places much greater value has been placed on inventing new knowledge. Even when invention is in good standing there can be a big question of who is to be permitted to do it. And even if invention is applauded it may be still true that this may only be in certain areas considered appropriate or important. This is as true in mathematics as in other areas of human activity.

In short, a lot of factors can shape what is seen as “mathematics”, what it is to be used for, and by whom. As an illustration it is worth remembering that astrology has until relatively recently been considered both a legitimate area of human knowledge and a key impetus for mathematical development. Thus E. G. Taylor writes of the understandings in England in the late sixteenth century: “The dictum that mathematics was evil for long cut at the very roots of the mathematical arts and practices. How were those to be answered for whom mathematics meant astronomy, astronomy meant astrology, astrology meant demonology, and demonology was demonstrably evil?”5 Indeed, it was noted that when the first mathematical Chairs were established at Oxford University, parents kept their sons from attending let they be ‘smutted with the Black Art’.6 However, despite these negative connotations, practioners of “the dark arts” played a strong role in developing and refining instruments and methodologies for recording and predicting the movement of “star signs” as they moved across the celestial sphere.

One of the key features of the contemporary world is its high level of interconnection. In such a world it is easy to imagine that developments in “mathematics” which happen in one place will be known and built on almost simultaneously in another. Yet that is a very modern concept. In most of history the movement of information across space and time has been slow and very imperfect. So what at what one time has been discovered in one place may well have been forgotten a generation or two later, and unheard of in many other places. For this reason, amongst others already mentioned, talk of the evolution of mathematics as if it had a definite timetable, and a single direction is likely to be very misleading.

History of course relies on evidence. We can only know where and when innovations have occurred when evidence of them can be uncovered. Even the partial picture thus uncovered reveals a patchwork of developments in different directions. That is certainly a shadow of the whole complex pattern of discovery, invention, forgetting, and re-discovery which will have been shaped at different times by particular needs and constraints of different cultures, values, political structures, religions, and practices. In short, understanding the evolution of calculating machines is assisted by investigating it in the context of the evolution of mathematical thinking. But that is no simple picture. The history of developments in calculators and mathematics has been embroidered and shaped by the the social, political and economic circumstances in which they emerged. At times, mathematical developments have shaped developments in calculators, and and other times, the opposite has been true.

3. What is a calculator?

“Calculator” could be taken to mean a variety of things. It could be calculation ‘app.’ on a smart phone, a stand alone elctronic calculator from the 1970s, or the motorised and before that hand-cranked mechanical devices that preceded the electronic machines. In earlier times it could simply mean someone who calculates. It is difficult to see where the line should be drawn in this regress all the way back to the abstract manipulation of ‘numbers’.

In this discussion, “calculator” is used as shorthand for “calculating technology”. In particular it is taken to mean any physically embodied methodology, however basic, used to assist the performance of arithmetic operations (including counting). Thus a set of stones laid out to show what the result is if two are added to three (to give five), or if in three identical rows of five what the outcome is of multiplying five by three (to give fifteen) will be regarded as a simple calculator. So too, will the fingers of the hand, when used for similar purpose, and even the marking of marks on a medium (such as sand, clay or papyrus) to achieve a similar result.

This approach is certainly not that taken in all the literature. Ernest Martin in his widely cited book The Calculating Machines (Die Rechenmaschinen) is at pains to argue of the abacus (as well as slide rules, and similar devices), that “it is erroneous to term this instrument a machine because it lacks the characteristics of a machine”.7 In deference to this what is referred to here is “calculators” (and sometimes “calculating technologies or “calculating devices”). Where the phrase “calculating machine” is used it will be in the sense used by Martin, referring to something with more than just a basic mechanism which would widely be understood to be a machine. But with that caveat, the term “calculator” will be used here very broadly.

The decision to apparently stretch the concept of calculator so far reflects a well known observation within the History and Philsophy of Science and Technology that in the end, technique and technology, or science and technology, are not completely distinct categories. Technologies embody knowledge, the development of technologies can press forward the boundaries of knowledge, and technological development is central to discovery in science. As Mayr says in one of many essays on the subject, “If we can make out boundaries at all between what we call science and technology, they are usually arbitrary.”8 Indeed, as will be described later, the mental image that mathematics is the work of mathematicians (‘thinkers’) whilst calculators are the work of artisans (‘practical working people’) is an attempt at a distinction that falls over historically, sociologically, and philosophically.

4. A discussion in three parts.

This is a work in progress, which in part is why it is formed as a website. So please regard it as a first draft (for which there may never be a final version). For this reason, corrections, additional insights, or links to other resources I should know about will be much appreciated.

A word also about the way I have constructed the historical account. In keeping with the analysis I have contributed to elsewhere (in a book by Joseph Camilleri and myself9), human development, will roughly be divided into a set of semi-distinct (but overlapping) epochs, preceded by a “pre-Modern Era” spanning the enormous time period from the birth of the first modern homo-sapiens to the beginning of the “Modern Period”. This beginning is set as beginning (somewhat earlier than is conventional) in the middle of the sixteenth century, with the “Early Modern Period” continuing from the mid-sixteenth to late eighteenth century, and the “Late Modern Period” stretching forward into the twentieth century, and terminating around the two world wars. From thereon the world is regarded by Joseph Camilleri and myself as entering a period of transition10 (but there is not much need to focus on that here).

Thus the historical account is broken into three parts. The first part looks at the relationship between the evolution of calculating and calculators in the pre-Modern period. That forms a backdrop but only one object in the collection is of an appropriate age. Apart from that object (which is some 4,000 years old) the objects in this “collection Calculant” are drawn from the Modern Period (the earliest of these objects being from the early sixteenth century), and the Late Modern Period (from 1800) when mechanical calculation began to gain greater use in the broader society.